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4-4: The Wave Nature of Electrons

As learned in Part 1 and 2, it has been made clear that, if a substance is divided into finer and finer pieces, we reach molecules and atoms, and then we realize that the atoms consist of electrons and nuclei. Namely, it has been clarified that matter is a collection of ultramicroscopic particles. In the classical physics up to the 19th century, these particles were considered to move obeying Newtonian mechanics and Maxwellian electromagnetism. However, this viewpoint has become doubtful after the proposal of the Bohr model of the atomic structure (Bohr's quantum theory).
On the other hand, light had been considered to be electromagnetic waves in the classical physics. However, after the discovery of light quanta (photons), it was clarified that the light has wave nature at one time and particle nature at another time. Namely, light has a kind of duality.
Thinking of this, L. V. de Broglie (France, 1892 - 1987) guessed that such substance particles as electrons or protons might have wave nature, although they had been considered to have particle nature so far.

[De Broglie Waves]
The idea of de Broglie waves or de Broglie matter waves is that, thinking of the fact that light which had been considered to be waves had the particle nature, such a substance particle as electron or proton which had been considered to be "particle" might have "wave nature".
Hence, de Broglie thought that, if the double aspect, i.e. the duality of the wave nature and the particle nature, is valid not only for light but also for such substance particles as electrons, Einstein's relations which connect the particle and wave aspects in light quanta

would be satisfied for de Broglie matter waves as well. Therefore the relations, Eq. (1), are often called Einstein - de Broglie's relations.
If we apply these relations to the case of the Bohr model of the hydrogen atom, we can well understand its plausibility as follows. If we consider that the electron in a hydrogen atom moves at constant speed along a circular orbit around the nucleus (proton), the quantum condition in Bohr's quantum theory is written as Eq. (3) on the preceding page. By using Einstein's relation in this equation, the quantum condition is written

This equation means that the circumference of the circular orbit of the electron must be a integral multiple of the wavelength of de Broglie wave. In other word, de Broglie wave accompanying the motion of the electron should be continuous. Therefore, we can easily understand the quantum condition that determines the stationary states by considering the continuity of de Broglie waves. (See the following figure.)
Bohr's quantum condition. The condition for stationary states
The circumference of the circular orbit of the electron should be an integral multiple of the wavelength of de Broglie wave, otherwise the wave cannot be smoothly continuous.

[The Laue Diagram -- Wave Nature of X-Rays]
In 1914, M. T. F. von Laue (Germany, 1879 - 1960) found a symmetrical pattern of spots on a photographic plate by a beam of X-rays that had passed through a crystal. This is called the Laue diagram, which is produced by interference of beams of X-rays diffracted on the different atomic planes in the crystal. It was confirmed by this phenomena that X-rays are electromagnetic waves of very short wavelengths.

The reason why the interference pattern in the Laue diagram appears is as follows: As shown in the above figure, an X-ray beam is irradiated on a crystal consisting of the atomic planes A, B, ... As light reflects on a mirror, so the X-ray beam striking the plane A with the incident angle reflects most strongly in the same angle . The similar happens on the plane B. These X-ray beams reflected on the planes A and B interfere with each other and constructive interference takes place when the difference in pathlength is equal to an integral number of wavelength , i.e.,

where d is the interplanar distance, and is the angle between the incident X-ray and the crystal plane. This is called Bragg's condition. Thus a bright spot will appear in the angle satisfying the condition. When the white X-rays which contain all wavelengths are illuminated on a crystal, various atomic planes in the crystal reflect the X-rays satisfying Bragg's condition selectively or exclusively, and then a symmetric pattern of spots is produced on the photographic plate placed behind. This is the Laue diagram, an example of which is shown in the following picture.
An example of the Laue diagram
This is a interference pattern of X-rays diffracted by a single crystal of silicon. The small black spots lined crosswise are the so-called Laue spots. Here, the wavelength and the interatomic distance are about (Courtesy of Prof. Y. Soejima, Dept. of Physics, Kyushu Univ.)

[The Empirical Evidences of the Wave Nature of Electrons]
The wavelength of de Broglie wave associated with the electrons accelerated by an electric potential of 100 V is about (). This is almost the same as that of ordinary X-rays. It is therefore supposed that, if we bombarded this electron beam on a crystal, we would observe a diffraction pattern similar to the Laue diagram in the case of X-rays.
In 1927, American physicists, C. J. Davisson (USA, 1881 - 1958) and L. H. Germer (USA, 1896 - 1971) observed an interference pattern of electron beams diffracted by a single crystal of nickel for the first time and, in the same year, G. P. Thomson (UK, 1892 - 1975) independently observed the diffraction pattern using a metallic multi-crystal. Next year, S. Kikuchi (Japan, 1902 - 74) was also successful in a similar experiment using a thin film of mica.
An example of the diffraction and interference phenomena of electrons is shown in the following picture.

A diffraction pattern of electron beams
Electron beams are diffracted by a crystal of manganese-nickel alloy. In this case, the de Broglie wavelength is shorter than which corresponds to rather high speed electron beam. (Courtesy of Prof. Y. Soejima, Dept. of Physics, Kyushu Univ.)

Looking at these results, people could no longer deny that electrons possess the double nature (dual property), i.e. the particle nature and the wave nature.
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